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Cosmological Constant and Torsion

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Cosmological Constant and Torsion Big Bounce and Dark Energy in Gravity with Torsion Nikodem J. Popławski Lecture Virtual Institute of Astroparticle Physics Paris, France 22 June 2012 Outline 1. Problems of standard cosmology 2. Einstein-Cartan-Sciama-Kibble theory of gravity 3. Dirac spinors in spacetime with torsion 4. Solution: cosmology with torsion • Nonsingular big bounce instead of singular big bang • Torsion as simplest alternative to inflation 5. Simplest affine theory of gravity • Cosmological constant from torsion E = mc2 Problems of standard cosmology • Big-bang singularity – can be solved by LQG But LQG has not been shown to reproduce GR in classical limit • Flatness and horizon problems – solved by inflation consistent with cosmological perturbations observed in CMB But: - Scalar field with a specific (slow-roll) potential needed fine-tuning problem not resolved - What physical field causes inflation? - What ends inflation? Existing alternatives to GR: • Dark energy • Use exotic fields • Are more complicated • Dark matter • Do not address all problems • Matter-antimatter asymmetry (usually 1, sometimes 2) Einstein-Cartan-Sciama-Kibble theory Spacetime with gravitational torsion This talk: Big-bang singularity, inflation and dark energy problems all naturally solved by torsion Affine connection • Vectors & tensors – under coordinate transformations behave like differentials and gradients & their products. • Differentiation of vectors in curved spacetime requires subtracting two vectors at two infinitesimally separated points with different transformation properties. • Parallel transport brings one vector to the origin of the other, so that their difference would make sense. dx A δA Affine connection E = mc2 Curvature and torsion Calculus in curved spacetime requires geometrical structure: affine connection Covariant derivative of a vector Two tensors constructed from affine connection: • Curvature tensor • Torsion tensor – antisymmetric part of connection É. Cartan (1921) Contortion tensor E = mc2 Theories of spacetime Special Relativity – flat spacetime (no affine connection) Dynamical variables: matter fields General Relativity – (curvature, no torsion) Dynamical variables: matter fields + metric tensor Degrees Connection restricted to be symmetric – ad hoc of freedom (equivalence principle) ECSK gravity (simplest theory with curvature & torsion) Dynamical variables: matter fields + metric + torsion E = mc2 ECSK gravity T. W. B. Kibble, J. Math. Phys. 2, 212 (1961) D. W. Sciama, Rev. Mod. Phys. 36, 463 (1964) Riemann-Cartan spacetime – metricity → Christoffel symbols of metric Matter Lagrangian density Total Lagrangian density like in GR: Two tensors describing matter: • Energy-momentum tensor • Spin tensor E = mc2 ECSK gravity Curvature tensor = Riemann tensor + tensor quadratic in torsion + total derivative Stationarity of action under → Einstein equations Stationarity of action under → Cartan equations • Torsion is proportional to spin density • Contributions to energy-momentum from spin are quadratic Dirac spinors with torsion Simplest case: minimal coupling Dirac Lagrangian density (natural units) Dirac equation Covariant derivative of a spinor GR covariant derivative of a spinor F. W. Hehl, P. von der Heyde, G. D. Kerlick & J. M. Nester, Rev. Mod. Phys. 48, 393 (1976) Dirac spinors with torsion Spin tensor is completely antisymmetric Torsion and contortion tensors are also antisymmetric LHS of Einstein equations comoving frame Fermion number density E = mc2 ECSK gravity Torsion significant when Uik » Tik (at Cartan density) 45 -3 For fermionic matter ½C > 10 kg m >> nuclear density Other existing fields do not generate torsion • Gravitational effects of torsion are negligible even for neutron stars (ECSK passes all tests of GR) • Torsion vanishes in vacuum → ECSK reduces to GR • Torsion is significant in very early Universe and black holes Imposing symmetric connection is unnecessary ECSK has less assumptions than GR E = mc2 Cosmology with torsion Spin corrections to energy-momentum act like a perfect fluid Friedman equations for a homogeneous and isotropic Universe: Statistical physics in early Universe (neglect k) E = mc2 Cosmology with torsion NP, Phys. Rev. D 85, 107502 (2012) Scale factor vs. temperature Solution reference values torsion correction Singularity avoided E = mc2 Cosmology with torsion Temperature vs. time → Can be integrated parametrically t ≤ 0 t ≥ 0 E = mc2 Cosmology with torsion Temperature vs. time Time scale of torsion era Cusp-like bounce E = mc2 Nonsingular big bounce instead of big bang Scale factor vs. time NP, Phys. Rev. D 85, 107502 (2012) Big bounce E = mc2 Nonsingular big bounce instead of big bang Scale factor vs. time slope discontinuity Big bounce E = mc2 Nonsingular big bounce Scale factor vs. time ? Mass generation Big bounce E = mc2 Nonsingular big bounce Singularity theorems? Spinor-torsion coupling enhances strong energy condition Expansion scalar (decreasing with time) in Raychaudhuri equation is discontinuous at the bounce, preventing it from decreasing to (reaching a singularity) E = mc2 Torsion as alternative to inflation For a closed Universe (k = 1): Velocity of the antipode relative to the origin At the bounce Density parameter Current values (WMAP) NP, Phys. Rev. D 85, 107502 (2012) E = mc2 Torsion as alternative to inflation Big bounce: Minimum scale factor Horizon problem solved Number of causally disconnected volumes Flatness problem solved No free parameters Cosmological perturbations – in progress E = mc2 Theories of spacetime General Relativity Dynamical variables: matter fields + metric tensor ECSK gravity Dynamical variables: matter fields + metric tensor + torsion Purely affine gravity (A. Eddington 1922, A. Einstein 1923, E. Schrödinger 1950) Dynamical variables: matter fields + affine connection • Metric tensor is constructed from matter Lagrangian & curvature • Field equations in vacuum generate cosmological constant • Field equations with matter are more complicated and differ from (physical) metric solutions E = mc2 Affine gravity Similar to gauge theories of other fundamental forces: • Affine connection (dynamical variable in affine gravity) generalizes an ordinary derivative to a coordinate-covariant derivative • Gauge potentials (dynamical variables in gauge theories) generalize an ordinary derivative to gauge-invariant derivatives E = mc2 Affine gravity Dynamical Lagrangian must contain derivatives of connection Simplest gravitational Lagrangian: linear in derivatives → linear in Ricci tensor (like in GR and ECSK) and contracted with an algebraic tensor constructed from connection (from torsion) Other Lagrangians based on NP, ArXiv:1203.0294 are unphysical E = mc2 Affine gravity Stationarity of action under → field equations Variation can be split into and For vacuum: → Christoffel symbols of tensor k Gravitational Lagrangian becomes Ricci tensor of tensor k Cosmological-like term E = mc2 Cosmological constant from torsion Defining Λ sets length scale of affine connection gives the Einstein-Hilbert action with cosmological constant Affine length scale Λ becomes cosmological constant Only configurations with are physical c, Λ, G – fundamental constants of classical physics (set units) Planck units set by h, c, G – their relation to Λ still unknown NP, ArXiv:1203.0294 E = mc2 Cosmological constant from torsion The metric in the matter Lagrangian must also be replaced by For ordinary matter (Dirac spinors, known gauge fields): same gravitational Lagrangian Total action sets mass units becomes the EH action with matter and Λ E = mc2 Cosmological constant from torsion If fields depend on torsion only through (spinors do not): → Equations in the presence of spinors – in progress Expected to reproduce or slightly modify ECSK with Λ 1/2 2 These modifications may contain Λ c ~ aMOND → dark matter Summary Torsion in the ECSK theory of gravity: • Averts the big-bang singularity, replacing it by a nonsingular, cusp-like big bounce • Solves the flatness and horizon problems without inflation Torsion in the simplest affine theory of gravity: • Gives field equations with a cosmological constant No free parameters .
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